Former SPD politician and Bundesbank manager Thilo Sarrazin has unleashed a torrent of outrage (but also support from some quarters) with his hypothesis that muslims are going to swamp Germany and make up the majority of the population within a century, simply by having more children. In spite of acres of coverage going over the subject matter back and forth, I haven't seen a mathematical treatment of the claim to check whether it could actually happen, so I did my own back of the envelope calculation.
What would it take to make Sarrazin's fears turn into reality? For a 10 fold population increase in a century, we need roughly a doubling per generation. Let's assume that traditionalist muslim families (henceforth: "traditionalists") have an average of 8 children, of whom 4 will follow the traditionalist way, have 8 children again, etc. With this model (key parameters: 50% uptake of traditionalist lifestyle, 8 children per tradionalist couple) the number of traditionalists would double from each generation to the next, so these (or higher) parameters would support Sarrazin's claims. Note that we can safely ignore the 50% of children who don't follow the traditionalist way, as they will gradually adjust their fertility to the prevailing level of the country which is sub-replacement level.
I haven't seen scientific data regarding these two numbers, but my gut feeling would be that both numbers will in fact be lower, so the traditionalist population will _not_ double in a generation, and may not even grow at all. And seeing that net immigration has vanished in recent years, we can ignore that part as well.
So here is my stationary model: Assuming that 1/3 of children from traditionalist households stick with the traditionalist ways and have an average of 6 children, that would leave us with just 2 traditionalists in the new generation to replace the 2 parents we started with, i.e. we have exact replacement.
Of course if the parameters are smaller than 1/3 and 6, the traditionalists will gradually disappear (though possibly not as quickly as the rest of the population).